Final answer:
To find the positive integers less than or equal to 100 divisible by 2 or 3 but not by 12, we count multiples of 2 and 3, subtract the multiples of both (6), and then exclude the multiples of 12. The final count is 59.
Step-by-step explanation:
To determine how many positive integers less than or equal to 100 are divisible by either 2 or 3, but not by 12, we need to count the number of multiples of 2 and 3 and then exclude the multiples of 12.
First, let's count the multiples of 2 (2, 4, 6, ..., 100). Since 2 is a divisor of 100, there are 100 / 2 = 50 multiples of 2.
Second, let's do the same for 3 (3, 6, 9, ..., 99). There are 99 / 3 = 33 multiples of 3.
However, some numbers are multiples of both 2 and 3, which makes them multiples of 6 (6, 12, 18, ..., 96). There are 96 / 6 = 16 such numbers.
Since we added these numbers twice, once in the count for 2 and once for 3, we need to subtract them once to correct our count. So far, the total count is 50 + 33 - 16 = 67.
Finally, we need to exclude multiples of 12, since the question asks us to exclude them. There are 100 / 12 = 8 multiples of 12 that we need to subtract from our total count.
Therefore, the final count of numbers that are divisible by either 2 or 3, but not by 12, is 67 - 8 = 59.